I understand that different forces can act as centripetal forces (shear, tension of a string etc) but in the case of the rotating earth, is it really the gravitational force the centripetal force that keeps the earth spinning? In that case, shouldn't the centripetal acceleration at the equator be equal to the gravitational acceleration of 9.81 m/s2, instead of the 0.03 m/s2 obtained using the linear velocity at the equator and the equatorial radius? What force actually acts as the centripetal force on earth or on any rotating sphere?
I have also read elsewhere that for someone standing on earth, the centripetal force is the net force resulting from the difference between the weight and the normal force. But where does this difference comes from (why is it different from 0)? As the gravitational attraction itself doesn't change, I believe this difference is due to a change in the normal force. But why does it change at all?